# Full text of "Treatise On Analysis Vol-Ii"

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```9   MEASURABLE FUNCTIONS       137

(13.9.8)    A function equivalent to a measurable function is measurable.

This follows from (13.9.4) and the fact that if K is compact and N c K
is negligible, then there exists a compact set K' c K — N with measure
arbitrarily close to the measure of K (13.7.9).

A mapping f defined almost everywhere in X, with values in a topological
space Y, is said to be measurable if every mapping of X into Y which is
equivalent to/ (13.6) is measurable. Clearly (13.9.8) it is enough for one such
mapping to be measurable.

(13.9.8.1) In particular, it follows, from (13.9.6) and (13.9.8) that if /and g
(with values in K) are defined and finite almost everywhere inX and measurable,
then/+ g and/# (which are defined almost everywhere) are measurable.

(13.9.9)    Let u be a measurable mapping 0/X into a topological space Y.IfM
is any open or closed subset ofY, then u^i(M) is measurable.

Consider a partition of X consisting of a sequence of compact sets Kn
and a negligible set N, such that the restrictions u \ Kn are continuous. If
M c Y is closed, then u~1(M) n Kn is compact (3.11.4), hence tT^M) is
measurable. If M is open, then X — w^M) = u~1(Y — M) is measurable,
and therefore so is u~*(M) (13.9.3).

(13.9.10) (Egoroff's theorem) Let Y be a metric space and let (/„) be a
sequence of measurable mappings ofX into Y such that, for almost all x e X,
the sequence (/,(*)) converges to a limit f(x). Then

( 1 )   fis measurable ;

(2) for each compact subset KofX and each s > 0, there exists a compact
subset K' of K such that ^(K - K') :g £ and such that the restrictions fn \ K'
are continuous and converge uniformly in K' tof\ K'.

Clearly it is sufficient to prove the second assertion ((13.9.4) and (7.2.1)).
Let K0 be a compact subset of K such that ju(K — K0) ^ le and such that the
restrictions/, | K0 are continuous for all n (13.9.5). If d denotes the distance
function on Y, let BM>r be the set of all x e K0 such that </(/,(*), /«(*)) £ 1/r
for at least one pair of integers (p, q) such that p 2> n and q ^ n. If Aptqtt, is
the set of points x e K0 such that d(fp(x)9fq(xj) ^ 1/r, then we havee restrictions u \ Ln is continuous.
```